Related papers: Identifiers for MRD-codes
We generalize the example of linear set presented by the last two authors in "Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$" (2019) to a more general family, proving that such linear sets are maximum scattered…
It is an important task to construct quantum maximum-distance-separable (MDS) codes with good parameters. In the present paper, we provide six new classes of q-ary quantum MDS codes by using generalized Reed-Solomon (GRS) codes and…
(Partial) Unit Memory ((P)UM) codes provide a powerful possibility to construct convolutional codes based on block codes in order to achieve a high decoding performance. In this contribution, a construction based on Gabidulin codes is…
In this paper we introduce a new class of extremal codes, namely the $i$-BMD codes. We show that for this family several of the invariants are determined by the parameters of the underlying code. We refine and extend the notion of an…
In this work we present results on the classification of $\mathbb{F}_{q^n}$-linear MRD codes of dimension three. In particular, using connections with certain algebraic varieties over finite fields, we provide non-existence results for MRD…
In this paper, we consider the Reed-Muller (RM) codes. For the first order RM code, we prove that it is unique in the sense that any linear code with the same length, dimension and minimum distance must be the first order RM code; For the…
A $q$-ary code of length $n$, size $M$, and minimum distance $d$ is called an $(n,M,d)_q$ code. An $(n,q^{k},n-k+1)_q$ code is called a maximum distance separable (MDS) code. In this work, some MDS codes over small alphabets are classified.…
We present a new general construction of MDS codes over a finite field $\mathbb{F}_q$. We describe two explicit subclasses which contain new MDS codes of length at least $q/2$ for all values of $q \ge 11$. Moreover, we show that most of the…
A class of scattered linearized polynomials covering infinitely many field extensions is exhibited. More precisely, the $q$-polynomial over $\mathbb F_{q^6}$, $q \equiv 1\pmod 4$ described in arXiv:1906.05611, arXiv:1910.02278 is…
We consider a new class of linear codes, called affine Grassmann codes. These can be viewed as a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. We determine the length, dimension, and the minimum…
Maximum distance separable (MDS) codes are optimal where the minimum distance cannot be improved for a given length and code size. Twisted Reed-Solomon codes over finite fields were introduced in 2017, which are generalization of…
The classical Reed-Muller codes over a finite field $\mathbb{F}_q$ are based on evaluations of $m$-variate polynomials of degree at most $d$ over a product set $U^m$, for some $d$ less than $|U|$. Because of their good distance properties,…
Sum-rank-metric codes have wide applications in universal error correction, multishot network coding, space-time coding and the construction of partial-MDS codes for repair in distributed storage. Fundamental properties of sum-rank-metric…
Maximum-distance separable (MDS) convolutional codes form an optimal family of convolutional codes, the study of which is of great importance. There are very few general algebraic constructions of MDS convolutional codes. In this paper, we…
Cyclic codes are the most studied subclass of linear codes and widely used in data storage and communication systems. Many cyclic codes have optimal parameters or the best parameters known. They are divided into simple-root cyclic codes and…
We define almost affine vector rank-metric codes as subsets $\mathcal{C}\subseteq \mathbb{F}_{q^m}^n$ whose canonical projections have cardinalities that are powers of $q^m$, and prove that they naturally induce $q$-matroids. We establish…
Let $\cal M$ denote the set ${\cal S}_{n, q}$ of $n \times n$ symmetric matrices with entries in ${\rm GF}(q)$ or the set ${\cal H}_{n, q^2}$ of $n \times n$ Hermitian matrices whose elements are in ${\rm GF}(q^2)$. Then $\cal M$ equipped…
The concept of linear set in projective spaces over finite fields was introduced by Lunardon in 1999 and it plays central roles in the study of blocking sets, semifields, rank-distance codes and etc. A linear set with the largest possible…
Rank-metric codes have been a central topic in coding theory due to their theoretical and practical significance, with applications in network coding, distributed storage, crisscross error correction, and post-quantum cryptography. Recent…
Given any linear code $C$ over a finite field $GF(q)$ we show how $C$ can be described in a transparent and geometrical way by using the associated Bruen-Silverman code. Then, specializing to the case of MDS codes we use our new approach to…